COMPOSITIONS OF MULTIVALUED FUNCTIONS JUN LE GOH Abstract. In reverse mathematics, one sometimes encounters proofs which invoke some theorem multiple times in series, or in- voke different theorems in series. One example is the standard proof that Ramsey's theorem for 2 colors implies Ramsey's the- orem for 3 colors. A natural question is whether such repeated applications are necessary. Questions like this can be studied un- der the framework of Weihrauch reducibility. For example, one can attempt to capture the notion of one multivalued function being uniformly reducible to multiple instances of another multivalued function in series. There are three known ways to formalize this notion: the compositional product, the reduction game, and the step product. We clarify the relationships between them by giving sufficient conditions for them to be equivalent. We also show that they are not equivalent in general. 1. Introduction Many mathematical theorems can be thought of as problems; that is, they have the form \for every instance X, there exists a solution Y ". For example, instances of the intermediate value theorem (on [0; 1]) are continuous functions f : [0; 1] ! R such that 0 lies strictly between f(0) and f(1), and solutions to f are zeroes of f. Another example is K¨onig'slemma, which states that every infinite finitely branching tree has an infinite path. Instances of K¨onig'slemma are infinite finitely branching trees T , and solutions to T are infinite paths on T . What would it mean to solve a problem? Given an instance of the problem, we must provide some solution to said instance. Since each instance of a problem may have many solutions, there may be many possible mappings which take each instance to a solution. Intuitively, Date: June 27, 2019. Key words and phrases. Weihrauch reducibility; compositional product; reduc- tion game; step product. We thank Richard A. Shore for many useful discussions and suggestions. We also thank the referees for their helpful comments. This research was partially supported by NSF grant DMS-1161175. 1 2 JUN LE GOH a problem is easily solvable if some such mapping can be constructed in a simple way. This gives us a way to study the computational content of theorems: we study how difficult it is to solve the associated problem. The ma- chinery to do the latter was first studied by computable analysts: for many theorems of interest, the instances and solutions of their associ- ated problems can be represented as elements of Baire space NN. The problems are then (possibly partial) multivalued functions on NN, and the realizers are single-valued functions on NN with the same domain. We are particularly interested in the relative computational strength of problems. Given any realizer for problem Q, can we computably transform it into some realizer for problem P ? In order to formalize this, we will use a reducibility relation known as Weihrauch reducibility. Our interest in comparing mathematical theorems up to Weihrauch reducibility is closely related to, and partially motivated by, the pro- gram of reverse mathematics, which studies the proof-theoretic strength of mathematical theorems over some base theory. The standard base theory is a weak subsystem of second-order arithmetic known as RCA0, which roughly corresponds to computable mathematics. In that con- text, one considers the following question. For any two theorems P and Q, do we have RCA0 + Q ` P ? For example, consider Ramsey's theorem for k-colorings of n-tuples n n (RTk ): for every coloring c :[N] ! k, there is an infinite c-homogeneous n n set. Then RCA0+RT3 ` RT2 (view the given 2-coloring as a 3-coloring). n This proof only invokes RT3 once, and it can be translated into a n n Weihrauch reduction from RT2 to RT3 . n n Less trivially, we also have that RCA0 + RT2 ` RT3 . The usual proof n n invokes RT2 twice, in series: given a 3-coloring of [N] by red, green, and blue, first define a 2-coloring of [N]n by red and \grue". Then use n RT2 to obtain an infinite homogeneous set for it. If we obtain a red homogeneous set, then we are done. If we obtain a \grue" homogeneous n set, then we apply RT2 to the original coloring restricted to this set, and we are done. n n 1 Is there a proof of RT3 which only invokes RT2 once? If not, is n n 2 there a proof of RT3 which invokes RT2 twice, but in parallel? We want to study such questions from the point of view of Weihrauch 1In the reverse mathematics setting, Hirst and Mummert [10] gave such a proof in RCA0. Their proof was not \uniform". In the setting of Weihrauch reducibil- ity, Hirschfeldt and Jockusch [9], Brattka and Rakotoniaina [5], and Patey [11] independently showed that there is no reduction. 2Note that invoking a theorem in parallel is a special case of invoking a theorem in series. COMPOSITIONS OF MULTIVALUED FUNCTIONS 3 reducibility. In order to do so, we must define some reducibility which would capture the notion of P being reducible to multiple instances of Q in series. There are three known ways to formalize this idea: (1) the compositional product (Definition 5); (2) reduction games (Definition 10); (3) the step product (Definition 18). In this paper, we clarify the relationships between these three notions (for example, Theorems 23, 27, Corollary 29). We conclude that they are (mostly) equivalent, and hence one is (mostly) free to use whichever definition is convenient for one's purposes. Along the way, we prove some basic properties of these notions, and give counterexamples where appropriate. We are also interested in capturing the notion of P being reducible to different theorems Q0;:::;Qn−1 in series. One motivating example is 2 Cholak, Jockusch, and Slaman's [6] proof of RT2 which proceeds by first using one theorem to obtain an infinite set on which the given coloring is stable, and then restricting to said set and obtaining, by another theorem, an infinite homogeneous set. To formalize this notion, we consider a generalized reduction game and show how it relates to the other formalizations (Theorem 34). In the rest of the introduction, we give some notation and basic def- initions. In this paper, P , Q, Q, etc., refer to multivalued functions from NN to NN. All multivalued functions and single-valued functions are allowed to be partial, unless otherwise stated. Their domains could be empty. Their domains and graphs need not be arithmetically defin- able, or even definable. If X is in the domain of P , then we say that X is a P -instance. If (X; Y ) 2 P , then we say that Y is a P -solution to X. If Φ is a Turing functional and X is an oracle for Φ, we will some- times write Φ(X) instead of ΦX . Since Φ formally only takes numbers as input, this should not cause confusion. We begin by defining Weihrauch reducibility on multivalued func- tions: Definition 1. For multivalued functions P and Q, we say that P is Weihrauch reducible (strongly Weihrauch reducible resp.) to Q, written P ≤W Q (P ≤sW Q resp.), if there is a forward functional Γ and a backward functional ∆ such that (1) for every P -instance X,ΓX is a Q-instance; (2) if X is a P -instance, then for every Q-solution Y to ΓX , ∆(X ⊕ Y ) (∆(Y ) resp.) is a P -solution to X. 4 JUN LE GOH Intuitively, P ≤W Q means that one can uniformly computably transform a realizer for Q into a realizer for P . In this paper, we focus on Weihrauch reducibility, but we use strong Weihrauch reducibility to state some of the results we need. Note the uniformity in the above definitions: Γ and ∆ have to sat- isfy the above conditions for all P -instances X. In fact, Weihrauch reducibility on multivalued functions was independently rediscovered by Dorais, Dzhafarov, Hirst, Mileti, and Shafer [7], who named it uni- form reducibility. See Brattka, Gherardi, and Marcone [3] for historical remarks about Weihrauch reducibility, and an equivalent definition. It is easy to see that ≤W is reflexive and transitive, so we can define the associated notion of Weihrauch equivalence and Weihrauch degrees: for multivalued functions P and Q, we say that P and Q are Weihrauch equivalent, written P ≡W Q, if P ≤W Q and Q ≤W P . For a multi- valued function P , its Weihrauch degree p is its ≡W -class. Weihrauch reducibility lifts to Weihrauch degrees in the usual way; that is, we say that p ≤W q if and only if there is some P 2 p and Q 2 q such that P ≤W Q, if and only if for all P 2 p and Q 2 q, we have P ≤W Q. We will abuse notation and use P ≤W q to mean that there is some Q 2 q such that P ≤W Q, or equivalently, for all Q 2 q, we have P ≤W Q. We give P ≡W q the analogous meaning. We can define strong Weihrauch equivalence and strong Weihrauch degrees in the same way. The Weihrauch degrees form a distributive lattice (Brattka, Gher- ardi [2], Pauly [12]). The join (coproduct) of multivalued functions P0 S and P1, denoted P0tP1, has instances i=0;1f(i; X): X is a Pi-instanceg. For i = 0; 1, (i; Y ) is a (P0 t P1)-solution to (i; X) if Y is a Pi-solution to X. The meet (sum) of P0 and P1, denoted P0 u P1, has instances f(X0;X1): Xi is a Pi-instanceg.